The period of a satellite moving in a circular orbit near the surface of a planet is independent of

General
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the period of a satellite moving in a circular orbit near the surface of a planet is independent of

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The period of a satellite moving in a circular orbit near the surface of a planet is independent of

Answer:
The period of a satellite moving in a circular orbit near the surface of a planet is a fascinating topic in orbital mechanics. To understand the independence factors, we need to break down the physical principles governing orbital motion.

When a satellite is in a circular orbit close to the surface of a planet, its period (T) can be derived from Newton’s law of gravitation and the centripetal force required to keep the satellite in orbit.

Solution By Steps:

  1. Expression for Orbital Period:

    • The gravitational force acting as the centripetal force:
      F_{\text{gravity}} = F_{\text{centripetal}}
      \frac{GMm}{r^2} = \frac{mv^2}{r}
      Here, ( G ) is the gravitational constant, ( M ) is the mass of the planet, ( m ) is the mass of the satellite, ( r ) is the orbital radius (which is approximately the radius of the planet if the satellite is close to the surface), and ( v ) is the orbital velocity of the satellite.

  2. Equating and Solving for Orbital Velocity:

    • Simplify and solve for the orbital velocity ( v ):
      v^2 = \frac{GM}{r}
      v = \sqrt{\frac{GM}{r}}

  3. Determine the Period Using the Orbital Circumference:

    • The period ( T ) is the time it takes for the satellite to make one complete orbit, which is the orbital circumference divided by the orbital velocity:
      T = \frac{2\pi r}{v}
      Substituting ( v ):
      T = \frac{2\pi r}{\sqrt{\frac{GM}{r}}}
      Simplifying gives:
      T = 2\pi \sqrt{\frac{r^3}{GM}}

  4. Near Surface Orbits:

    • If the satellite orbits very close to the planet’s surface, ( r \approx R ) (where ( R ) is the radius of the planet):
      T_{\text{surface}} = 2\pi \sqrt{\frac{R^3}{GM}}

  5. Independence Factors:

    • This equation shows that the period ( T ) depends on ( R ) (radius of the planet) and ( M ) (mass of the planet) along with the gravitational constant ( G ).
    • Notice that the satellite’s period is independent of the satellite’s mass ( m ).

Therefore, the period of a satellite moving in a circular orbit near the surface of a planet is independent of the satellite’s mass, atmospheric conditions, and the altitude above the surface, given the altitude is relatively negligible compared to the planet’s radius. The determining factors are the planet’s mass and radius, which dictate the gravitational field strength.